An upper bound for the first nonzero Steklov eigenvalue

¹ Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS 67260, USA
² School of Mathematical Sciences, Soochow University, Suzhou 215006, PR China
³ School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

^* Corresponding author: kuiwang@suda.edu.cn

Received: 24 January 2024
Accepted: 29 November 2024

Abstract

Let (Mⁿ, g) be a complete simply connected n-dimensional Riemannian manifold with curvature bounds Sect_g ≤ κ for κ ≤ 0 and Ric_g ≥ (n − 1)Kg for K ≤ 0. We prove that for any bounded domain Ω ⊂ Mⁿ with diameter d and Lipschitz boundary, if Ω* is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ₁(Ω) ≤ Cσ₁(Ω*), where σ₁(Ω) and σ₁(Ω*) denote the first nonzero Steklov eigenvalues of Ω and Ω* respectively, and C = C(n, κ, K, d) is an explicit constant. When κ = K, we have C = 1 and recover the Brock–Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space.